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電子課本網(wǎng) 第123頁

第123頁

信息發(fā)布者:
$證明:∵四邊形???ABCD???為矩形$
$??? ∴∠B=∠C=90°,??????AB=CD???$
$??? ∵EF⊥ED ???$
$??? ∴∠BEF+∠CED=∠FED=90°???$
$??? ∵∠BEF+∠BFE=90°???$
$??? ∴∠BFE=∠CED???$
$ 在???△BFE???和???△CED???中$
$??? \begin{cases}∠B=∠C\\∠BFE=∠CED\\EF=ED\end{cases}???$
$??? ∴△BFE≌△CED(\mathrm {AAS})???$
$??? ∴BE=CD???$
$??? ∴AB=BE,???$
$??? ∴∠BAE=45°???$
$??? ∴AE???平分???∠BAD???$

$證明:???(1)???由翻折的性質(zhì)可得,???∠ABE=∠EBM=\frac 12∠ABM,??????∠BDF=∠CDF=\frac 12∠BDC???$
$??? ∵AB//CD???$
$??? ∴∠ABD=∠BDC???$
$??? ∴∠EBM=∠BDF???$
$??? ∴BE//DF???$
$??? ∵AD//BC???$
$ ∴四邊形???BFDE???為平行四邊形$
$??? (2)∵???四邊形???BFDE???為菱形$
$??? ∴BF=DF???$
$??? ∴∠FBD=∠BDF???$
$??? ∵∠FBD+2∠BDF=90°???$
$??? ∴∠FBD=30°???$
$??? ∵AB=CD=2???$
$??? ∴BD=4?$
$?∴AD=\sqrt{42-22}=2\sqrt{3}???$
$?∴BC=AD=2\sqrt{3}.?$